Spectral growth conjecture for Salem numbers in $W(E_{10})$

Let $W(E_{10})$ be the Coxeter group under consideration, and let $\rho(d4e)$ denote the spectral radius of an element $d4e\in W(E_{10})$. For each level $n$, let $\Lambda_n\subset\mathbb{R}_{>1}$ be the set of spectral radii arising at $s_0$-level $n$, let $M_n$ and $m_n$ denote its maximal and minimal extreme values, and let $\hat\Lambda_n$ be the subset of primitive Salem numbers in $\Lambda_n$. Spectral Growth Conjecture. The maximal values satisfy

$$M_n=M_1^n,\qquad n\geq 1,$$

while the minimal extreme values approximately satisfy

$$m_n\approx\delta^{n-2}M_{n-1},\qquad n\geq 3,$$

for some $0<\delta<1$. Moreover,

$$|\hat\Lambda_n|\sim C\cdot r^n,$$

with $r\approx 5$, and the maximal gap between consecutive elements of $\hat\Lambda_n$ decays exponentially, so that the primitive spectrum becomes increasingly dense as $n\to\infty$. The approximation symbol indicates numerical proximity. These claims describe the experimentally observed spectral hierarchy in $W(E_{10})$: the exact formula for $M_n$ is proved in the source, while the scaling, exponential counting law, and density assertion remain conjectural.